A sampling theorem for space-variant systemsRobert J. Marks II, John F. Walkup, and Marion 0. Hagler

Department of Electrical Engineering, Texas Tech University, Lubbock, Texas 79409(Received 23 February 1976)

A sampling theorem applicable to that class of linear systems characterized by sufficiently slowly varying line-spread functions is developed. For band-limited inputs such systems can be exactly characterized withknowledge of the sampled system line-spread function and the corresponding sampled input. The desiredsampling rate is shown to be determined by both the system and the input. The corresponding output isshown to be band limited. A discrete matrix representation of the specific system class is also presented.Applications to digital processing and coherent space-variant system representation are suggested.

INTRODUCTION

This paper presents a sampling theorem applicable tothat class of linear space-variant systems characterizedby sufficiently slowly varying line-spread functions.For band-limited inputs, such systems can be exactlycharacterized with knowledge of the sampled systemline-spread function and the corresponding sampled in-put. The resulting sampling theorem expression for the(band-limited) system output is simply a summation ofconvolutions. A discrete matrix representation of thespecific system class is also presented.

Areas of possible application for the result includedigital signal processing and the representation of co-herent space-variant systems. Application limitationsare also briefly discussed.

Previous work in this area has been limited to sam-pling theorem expansion of the system line-spread func-tionl1 2 without regard to the input. Huang3 has discussedthe minimum required sampling rates taking the inputinto account.

For clarity of presentation, and without loss of gener-ality, attention will here be restricted to one dimension.Generalization to two or more dimensions may be ac-complished by straightforward extension.

SAMPLING THEOREM

The output g(x) of a linear system for a correspondinginput f(x) may be expressed via the superposition inte-gral

g(x) = f (4) h(x - 4; 4) d4 , (1)

= f (4) $x [h(x; 4) ] exp (- j2irfx ) d4

=$ zxjf Q) h(x;4)jv=f, ' (3)

where v andf, are the frequency variables associated,respectively, with 4 and x, and where, for a given two-dimensional function p(x;4), the Fourier transform op-erators are defined as

&,x[ p(x; t) I J p (x;t) e xp (- j 27rfffx)dx

and

5r [ p(x;4)li f (x;,)exp(-ij2Tvv)d4= .,

(4)

(5)

Roughly, j(-) operates on the input variable 4, whileS,(-) operates on the output variable x.

We now define the system's spatial transfer functionas

H.,(fx; t)A&X[h(x;4)]

Equation (3) may now be written

G(fz) =Stf(4)Hgx (fix;4A)] x.

(6)

(7)

In a similar fashion, we define the system's variationspectrum as

H,(x;v)!_ $ h(x;4 )]

The variation spectrum is a measure of how the line-spread function varies with changing 4. We say theline- spread function varies sufficiently slowly if thevariation spectrum is band limited5 in v for all x:

H,(x;v) = 0 for I v i > WV for all x

(8)

where h(x - 4; 4), the system line-spread function, isthe system response to an input Dirac delta located atthe point x = 4 (after the notation of Lohmann and P aris 4).

When the line-spread function is no longer a functionof its second argument, the system is isoplanatic (spaceinvariant), and Eq. (1) becomes the convolution integral

g(x)= f f ()h(x- 4)d4 =f(x)*h(x) ' (2)

The direct statement of a space-variant system's out-put spectrum may be found through application of Fou-rier transform operators to the superposition integral[Eq. (1)]:

G(f) =$[g(x)]

918 J. opt. Soc. Am., Vol. 66, No. 9, September 1976

(9)

The bandwidth 2W, is appropriately termed the varia-tion bandwidth. Note than an isoplanatic system has avariation bandwidth of zero, and is thus truly "invari-ant. "

Consider, then, the following form of the superposi-tion integral's integrand contained in Eq. (3):

f (4) h(x; 4) (10)

Multiplication in the space (4) domain corresponds toconvolution in the frequency (v) domain. As such, iff(4) and h(x;4) have respective bandwidths in v of 2Wfand 2W14, then their product will have a bandwidth 2Wequal to the sum of the component bandwidths:

2W=2Wf +2W,, . (11)

Copyright © 1976 by the Optical Society of America

FIG. 1. Generation of a sample line-spread function and cor-responding sample-transfer function for an arbitrary coherentspace-variant system.

One may then apply theWhittaker-Shannon samplingtheorem 6 to the product integrand to give

f(4)h(x;4)= f (4) h(x; ta) sinc2W(4-. ) - (12)

where 4, =n/2W and sincy = Sirectx] where

rect(x){ X I (13)

Substituting Eq. (12) into Eq. (3) followed by simplifi-cation leaves

G(f) 4 Zf 4) H (Ax;4)

x exp(- j27fx An) rect(fw) , (14)

or equivalently, in the space domain,

g(x) =Zf (4,) h(x - 4~; 4) * sinc(2Wx) . (15)n

Thus, providing that h(x;4) and f () are band limited in4, the output to a linear space-variant system can becomputed by (i) sampling the input, (ii) multiplying eachinput sample by its corresponding line-spread function,(iii) summing these products, and (iv) passing the sumthrough a suitable low-pass filter.

APPLICATION

It has previously been suggested that multielement co-herent space-variant systems may be represented by anumber of sample transfer functions. 7 '10 The vaststorage capacity of the volume hologram" may be uti-lized for sequential angle-multiplexed recording ofthese sample functions. The resulting volume hologramshould exhibit the input-output relationship of the orig-inal system to a good approximation. Such a systemrepresentation provides for increased orientation sta-bility, reduced weight, and real-space condensation.

For coherent optical systems, a sample transferfunction can easily be realized as in Fig. 1.7 8,10 Theimpulse input to the system is generated by focusingan incident plane wave to a line source at the input co-ordinate 4 =a. The corresponding line-spread functionh(x - a; a) is Fourier transformed by a displaced thinlens to yield in the back focal plane an amplitude dis-tribution proportional to a scaled and shifted versionof the sample spatial transfer function Hj(fX - a/xf;a).The scaled spatial frequency is given by

where A is the wavelength of the spatially coherent il-lumination and f is the focal length of the Fourier trans-forming lens. The amplitude and phase of a number ofsuch sample transfer functions may then be angle mul-tiplexed within a single volume hologram. The holo-gram, in principle, may then be utilized as a space-variant equivalent to a Vander Lugt filter.6 Suchschemes have been proposed by Deen, Walkup, andHagler, 7 and by Marks. 10 The method of Deen et al.falls short of direct implementation of the samplingtheorem [Eq. (14)] only by not including the requiredlow-pass filter of bandwidth 2W.

In practice, one is of course limited to recording onlya finite number of holograms, short of the countablyinfinite number required by the sampling theorem. Weovercome this problem by application of the familiarspace-bandwidth product estimate of the number of sam-ples required for a good approximation. If the systeminputf (4) is essentially zero outside of the interval I 41'a, and the spectrum (in v) of the integrandf(4)h(x;4)is essentially zero outside of the interval I v I : W, thenthe required number of samples for a good approxima-tion is'2

S =4Wa (17)

Truncation will of course result in a degree of error. 13

EXAMPLE

As an example application of the space-variant sys-tem sampling theorem, consider the ideal band-limitedcoherent imaging system with magnification M •1.While a simple mathematical coordinate transformationreduces the imaging system to an isoplanatic system, 6the ideal imaging system having nonunity magnificationmust rigorously be classified as space variant. 14 Thatis, in the physical sense, one may not use a singleplanar holographic filter in the Fourier plane to repre-sent a simple magnifier.

The line- spread function of the imaging system to beconsidered is

h(x - t;4) =2f, sinc2f0[x - M4]

=2fosinc2f(x- 4)- (MW-1)4 ], (18)

where fo is the cutoff frequency of the system. Notethat for an arbitraily large value of fM, Eq. (18) ap-proaches the displaced Dirac delta function character-izing an ideal imaging system. From Eq. (18) we maywrite

h(x;4)=2f 0 sinc2f 0[x- (M- 1)4l] (19)

The spatial transfer function [Eq. (6)] is then given by

H,(A;4) =exp[-j27r(M- 1)A4f]rect(fx/2f0 ) (20)

Substituting into Eq. (7) yields the system output spec-trum

G(fr) = F(Mf,) rect(f1/2f0 )

where

(21)

(22)

(16) Contained in Eq. (21) is the inherent low-pass nature

919 J. Opt. Soc. Am., Vol. 66, No. 9, September 1976

f. =x/xf,

Marks et al. 919

of the imaging system.

To apply the sampling theorem, we need first look atthe system's variation spectrum. Appropriately trans-forming Eq. (19) gives

He(x;v) =M 1 exp[ j27Tv(UX l)] rect( 2 f) 1))(23)

The finite system variation bandwidth is thus

2Wv =2f 0JM-1J . (24)

Note that for M=1, the line-spread function of Eq. (18)is isoplanatic and the corresponding variation band-width is zero.

Since the variation bandwidth of Eq. (23) is finite,the sampling theorem is directly applicable. Supposean input of bandwidth 2Wf is sampled at the rate 2W=2Wf +2Wv. The corresponding sampled expansion forthe output spectrum [Eq. (14)] becomes

G(A) = Zf (Q) exp(-j27Tf. M4n) rect(.{i j rect.),

(25)This relationship is recognized as a Fourier series ex-pansion of the output spectrum of Eq. (21) with period2W. The rect(x/2W) term merely retains the desiredzero-order term.

MATRIX REPRESENTATION

The space-variant sampling theorem results can beutilized to express the system input-output relationshipin exact matrix form. Such a relationship would finduse in digital applications.

Consider first the output spectrum expansion of Eq.(14). The rect(fx/2W) term dictates that a band-limitedinput to a space-variant system with finite variationbandwidth must result in an output with bandwidth notexceeding 2W. The output spectrum nmy thus be ex-pressed by the Whittaker-Shannon sampling theorem as

G(fA) =Z1 g(x) exp(-j27fx,,x) rect(U4 , (26)

where

x, =n/2W (27)

From this expansion we will obtain the desired outputsample values given by g(x,). Equating Eq. (26) withEq. (14) and multiplying both sides by exp(j27Tf,, ,,) gives

Eg(xn) exp(- j2f,(n - im)) rect({,)

=I~f(4,)H(fx;4n)exp(Q2If7 ( ))rect(.ff) (28)

We define the low-pass filtered sample transfer func-tion as

1 ,(f.; tn)=Hx(f.; n)rect(fx/2W) (29)

and recognize that

fep(j27rf,(n /- L2WUJexp~ 2W )rect(4f) df.= 2Wsinc(n - m) = 2W6n,, (30)

920 J. Opt. Soc. Am., Vol. 66, No. 9, September 1976

where n,,m is the Kronecker delta. Thus, integration ofEq. (28) over all fx gives

g(xm )mw~ f(4n) (Xm = (31)

where h(x; 4,) and l.(f=; en) are Fourier transform pairs.This relationship can be viewed as an infinite matrixrepresentation of the superposition integral [Eq. (1)1.Coupled with the space-bandwidth product [Eq. (17)] asa measure of the number of required samples, such arelationship would appear to have interesting applica-tions in digital signal processing.

CONCLUSION

Linear space-variant systems with line-spread func-tions of finite variation bandwidth may be representedexactly in sampled form for band-limited inputs. Em-ploying a sampling rate equal to the sum of the inputand variation bandwidths yields a relationship in whicheach sampled input point is assigned a correspondingline-spread function. This result gives further credibil-ity to the concept of holographic representation of lin-ear space-variant systems with volume holograms. Thecorresponding exact matrix characterization of thespace-variant system input-output relationship has anal-ogous applications in digital signal processing.

ACKNOWLEDGM ENTS

The authors express their appreciation to Dr. ThomasF. Krile of the Department of Electrical Engineering atRose-Hulman Institute of Technology in Terre Haute,Indiana for his helpful comments during the formulationof this paper. This research was supported by the AirForce Office of Scientific Research, Air Force SystemsCommand, 'USAF, under Grant No. AFOSR-75-2855A.

APPENDIX

The sampling theorem expressions in Eqs. (14) and(15) are not optimum in the sense of utilizing maximumallowable sampling intervals. That is, we are samplingboth the input and line-spread function at a rate of 2W,while the minimum required sampling rates are 2Wfand 2W,, respectively. As will be shown, however,the resulting expression employing these minimum sam-pling rates is rather unattractive for computation andimplementation purposes.

Consider, then, the following sampling theorem ex-pansion of a space-variant system's line-spread func-tion':

h(x; 4) =E h(x; 4) sinc2W( - p) ,p

(Al)

where 2W, is the variation bandwidth and 4p =p/2W,.One may similarly apply the sampling theorem to thesystem input to give

f (4) =Zf (Q) sinc2W1 (4 - k),k

where 4 =k/2Wf and 2Wf is the input's bandwidth.stituting Eqs. (Al) and (A2) into Eq. (3) gives

(A2)

Sub-

Marks et al. 920

(31)

GUY = 4W-W- EZ f (Q) EHx(x; 4)[rect(4)

> exp(-j27wf )] * [rect(jjW )erp(-j2vf% ¾)](A

Equation (A3) is identical to Eq. (14) yet employs largersampling intervals. The above relationship, however,has the disadvantage of not assigning each sample inputvalue to a single corresponding sampled line-spreadfunction.

Lastly, note that the two convolving rect' s in Eq.(A3) give an upper bound on the output bandwidth of 2W=2Wf +2W. This constraint is the same as containedin Eq. (14).

T. Kailath, "Channel Characterization: Time-Variant Disper-sive Channels, " in Lectures on Communications SystemTheory, edited by E. J. Baghdady (McGraw-Hill, New York,1960), pp. 95-124.

2N. Liskov, "Analytical Techniques for Linear Time-VaryingSystems," Ph.D. dissertation (Electrical Engineering Re-search Laboratory, Cornell University, Ithaca, N. Y. 1964)(unpublished), pp. 31-52.

3T. S. Huang, "Digital Computer Analysis of Linear Shift-Vari-ant Systems, " in Proc. NASA/ERA Seminar December,1969 (unpublished), pp. 83-87.

4A. W. Lohmann and D. P. Paris, "Space-Variant Image For-

mation, " J. Opt. Soc. Am. 55, 1007-1013 (1965).5Here, and in the material to follow, 'band limited" refers

specifically to that case where the spectrum is nonzero onlyover a single interval centered about zero frequency. It ap-pears, however, that the results can be extended to any spec-trum with finite support by application of corresponding sam-pling theorems. For example, see D. A. Linden, "A Discus-sion of Sampling Theorems, " Proc. IRE 47, 1219-1226(1959).

6 J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1968).

7 L. M. Deen, J. F. Walkup, and M. 0. Hagler, "Representa-tions of Space-Variant Optical Systems Using Volume Holo-grams," Appl. Opt. 14, 2438-2446 (1975).

8 L. M. Deen, "Holographic Representations of Optical Sys-tems, " M. S. thesis (Department of Electrical Engineering,Texas Tech University, Lubbock, Tex., 1975) (unpublished),pp. 37-60.

9 R. J. Marks II and T. F. Krile, "Holographic Representa-tion of Space-Variant Systems; System Theory," to appearin Appl. Opt.

10R. J. Marks 1I, "Holographic Recording of Optical Space-Vari-ant Systems," M. S. thesis (Rose-Hulman Institute of Tech-nology, Terre Haute. Ind., 1973) (unpublished), pp. 74-93.

"R. J. Collier, C. B. Burckhardt, and L. H. Lin, OpticalHolograpy (Academic, New York/London, 1971), pp. 466-467.

12D. Slepian, "On Bandwidth, " Proc. IEEE 64, 292 (1976).13K. Yao and J. B. Thomas, 'sOn Truncation Error Bounds for

Sampling Representations of Band-Limited Signals, "' IEEETrans. Aerospace Electron. Syst. 2, 640-647 (1966).

14A. A. Sawchuk, "Space-Variant Restoration by CoordinateTransformation," J. Opt. Soc. Am. 64, 138-144 (1974).

Copyright © 1976 by the Optical Society of America 921921 J. Opt. Soc. Am., Vol. 66, No. 9, September 1976